Section 3.1 Functions
#1 - 6: Which of the relations define y as a function of x?
1) { (1,2) (3,2) (4,2) (5,2)}
2) { (6,1) (7,1) (8,1)}
3) { (1,2) (3,4) (5,6) (7,8) (9,10)}
4) { (1,2) (4,5)}
5) { (3,1) (4,5) (3,6) }
6) { (3,7) (1,5) (1,2) }
#7 - 12: Use the vertical line test to determine whether the relation defines y as a function of x.
7)
8)
y
y
6
6
5
5
4
4
3
3
2
2
1
1
x
x
-6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-5
-4
-3
-2
-1
1
2
3
4
5
6
6
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
9)
10)
y
y
6
6
5
5
4
4
3
3
2
2
1
1
x
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-6
-5
-4
-3
-2
-1
1
6
2
3
4
5
3
4
6
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
11)
12)
y
y
6
6
5
5
4
4
3
3
2
2
1
1
x
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-6
-5
-4
-3
-2
-1
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
2
5
6
Section 3.1 functions
#13 – 26: Determine whether the equation defines y as a function of x. Hint, solve the equation for y
and sketch a graph using your calculator, then solve like problems #7-12.
3
5
13) y = x2
14) y = x2 +4
15) 𝑦 = 𝑥
17) 𝑦 = √𝑥 + 2
18) 𝑦 = √𝑥 − 2
19) 𝑦 = √𝑥 − 2
20) 𝑦 = √𝑥 − 5
21) y2 +x2 = 9
22) (x-2)2 + y2 = 16
23) x = y2
24) x + 2 = y2
25) 2x2 + 4y2 = 16
26) 5x2 + 3y2 = 25
16) 𝑦 = 𝑥
3
3
#27 – 45: Determine the domain and range of each function, write your answer in interval notation
when appropriate.
27) f = { 1,2) (3,4) (5,6)}
28) g = {(3,2) (5,7) (9,1) (8,1)}
29) g = {(0,4) (1,5) (2,6) (3,8) (4,1)
30) f = { (0,3) (-1,5)}
31)
32)
y
y
14
6
13
12
5
(1,12)
4
9
3
11
10
(3,4)
8
2
7
6
1
5
4
-6
-5
-4
-3
-2
3
-1
(1,2)
1
2
-1
2
-2
1
-6
-5
-4
(-5,0)
-3
-2
-1
x
1
-1
-2
-3
2
3
4
5
6
(-3,-2)
-3
(1,-3)
-4
-5
-4
(-3,-4)
-5
-6
-7
-6
x
3
4
5
6
Section 3.1 functions
32)
33)
14
y
12
(1,12)
10
10
(0,10)
8
8
6
6
4
-6
-4
4
(5,5)
(3,1)
2
-8
-2
2
-2
4
2
x
x
6
-8
8
-6
-4
-2
(-5,0)
(-3,-4)
-4
-6
2
4
6
8
-2
-4
-6
-8
-8
-10
-10
-12
-12
-14
-14
34)
35)
y
y
6
6
5
5
4
4
(2,4)
3
3
2
2
x
-5
-4
-3
-2
-1
1
(0,0)
(-1,-2)
-1
(-1,2)
(0,0)
1
1
-6
y
14
12
2
3
4
5
6
-6
-5
-4
-3
-2
-1
1
x
2
3
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
(2,-4)
4
5
6
36)
37)
y
y
6
6
5
5
4
4
3
3
2
2
(2,1)
1
1
(2,1)
-6
-5
-4
-3
-2
-1
1
x
2
3
4
5
x
6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
38)
39)
y
y
6
6
5
5
4
4
3
3
2
2
1
1
x
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-6
-5
-4
-3
-2
-1
2
3
4
5
-1
(2,0)
-1
1
(0,0)
6
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
40)
41)
y
y
6
6
5
5
4
4
3
3
2
2
(3,1)
(3,1)
1
1
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
2
3
4
5
6
6
Section 3.1 functions
42)
43)
45)
44)
Section 3.1 functions
#46 – 67 Use algebra to find the domain of each function. Write your answer in interval notation.
46) 𝑓(𝑥) = √𝑥 − 2
47) 𝑓(𝑥) = √𝑥 − 3
49) 𝑔(𝑥) = √2𝑥 + 10
50) 𝑓(𝑥) = 𝑥−3
52) 𝑓(𝑥) =
55) 𝑓(𝑥) =
58) 𝑔(𝑥) =
2
𝑥 2 +6𝑥−7
3
√𝑥−5
4
√6−𝑥
𝑥+2
53) 𝑔(𝑥) =
5
𝑥 2 −5𝑥+6
56) 𝑔(𝑥) = √5 − 𝑥
59) 𝑓(𝑥) =
2
√3−𝑥
48) 𝑔(𝑥) = √3𝑥 + 12
𝑥−6
51) 𝑓(𝑥) = 𝑥−7
54) 𝑓(𝑥) =
2
√𝑥−3
57) 𝑘(𝑥) = √6 − 𝑥
60) f(x) = 3x + 6
61) g(x) = 2x - 10
62) f(x) = x2 + 4
63) g(x) = x2 + 5
64) h(x) = (x-3)2 + 1
65) f(x) = (x+2)2 – 5
66) f(x) = x3 -6x2 + 7
67) g(x) = x3 – 4x2 + 2x – 3
Section 3.2 operations with functions
#1 - 18: Consider the functions defined by and find the requested function values.
2
𝑘(𝑥) =
𝑥+3
f(x) = 3x + 4
g(x) = x2 + 5x + 6
h(x) = 4
1) f(3)
2) f(-2)
3) g(1)
4) g(0)
5) h(2)
6) h(3)
7) k(-5)
8) k(-6)
9) f(b)
10) f(c)
11) f(b+1)
12) f(b-2)
13) g(2a)
14) g(3a)
15) g(x-2)
16) g(x+1)
17) k(a)
18) k(a-2)
#19 - 27: Let f(x) = 2x + 3 and g(x) = 2x2 + 5x + 3. Find each function.
19) (f + g)(x)
20) (g – f)(x)
21) (f/g)(x)
22) (𝑔 ∙ 𝑓)(𝑥)
23) ( g/f)(x)
24) (𝑓 ∘ 𝑔)(𝑥)
25) (𝑔 ∘ 𝑓)(𝑥)
26) (g+f)(x)
27) ( f – g)(x)
#28 - 36: Let f(x) = 2x2 – 5x – 3 and g(x) = x-3. Find each function.
28) (f + g)(x)
29) (g – f)(x)
30) (f/g)(x)
31) (𝑔 ∙ 𝑓)(𝑥)
32) (g/f)(x)
33) (𝑓 ∘ 𝑔)(𝑥)
34) (𝑔 ∘ 𝑓)(𝑥)
35) (g+f)(x)
36) ( f – g)(x)
Section 3.2 operations with functions
#37 - 48: Let h(x) = x2 + 2x+ 1 and k(x) = 2x - 5. Find each of the following.
37) (h+k)(3)
38) (hk)(-1)
39) (h/k)(5)
40) (k– h)(0)
41) (h– k)(7)
42) (kh)(4)
43) (ℎ ∘ 𝑘)(4)
44) (ℎ ∘ 𝑘)(0)
45) (𝑘 ∘ ℎ)(3)
46) (ℎ ∘ 𝑘)(−2)
47) (𝑘 ∘ ℎ)(1)
48) (𝑘 ∘ ℎ)(−6)
#49 – 57: Let s(x) = x2 +5x – 3 and t(x) = 2x - 7. Find each of the following.
49) (s/t)(3)
50) (s– t)(4)
51) (t+s)(6)
52) (𝑠 ∘ 𝑡)(4)
53) (𝑠 ∘ 𝑡)(0)
54) (𝑠 ∘ 𝑡)(3)
55) (𝑠 ∘ 𝑡)(−2)
56) (𝑡 ∘ 𝑠)(1)
57) (𝑡 ∘ 𝑠)(−6)
#58-67: Find the difference quotient; that is find
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
58) f(x) = 2x + 6
59) f(x) = 3x – 7
60) f(x) = 5x + 4
61) f(x) = 9x - 5
62) f(x) = x2 – 5
63) f(x) = x2 + 1
64) f(x) = x2 + 3x+ 5
65) f(x) = x2 + 5x – 3
66) f(x) = x2 – 2x + 4
67) f(x) = x2 – 5x + 8
Section 3.3 the graphs of functions
1) We will refer to the function in the graph as f.
a) find the x-intercepts
b) find the y-intercept
c) for what values of x is f(x) = 5
d) find f(5)
e) what is the domain of f
f) what is the range of f
2) We will refer to the function in the graph as g.
y
14
13
12
11
10
(6,7)
9
8
7
6
5
4
3
(-1,0)
(5,0)
2
1
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
-1
-2
(0,-5)
-3
-4
-5
-6
-7
-8
-9
a) find the x-intercepts
b) find the y-intercept
c) for what values of x is g(x) = 7
d) find g(0)
e) what is the domain of g
f) what is the range of g
(2,-9)
4
5
6
x
7
8
9
Section 3.3 the graphs of functions
3) We will refer to the function in the graph as h.
y
14
13
12
11
10
9
(-1,9)
(0,8)8
7
6
(-3,5)
(1,5)
5
4
3
2
1
-9
-8
-7
-6
(-4,0)
-5
-4
-3
-2
(2,0)
1
2
-1
x
3
4
-1
-2
-3
-4
-5
-6
(-5,-7)
-7
-8
-9
a) find the x-intercepts
b) find the y-intercept
c) for what values of x is h(x) = 0
d) find h(0)
e) what is the domain of h
f) what is the range of h
4) We will refer to the function in the graph as f.
a) find the x-intercepts
b) find the y-intercept
c) for what values of x is f(x) = 9
d) find f(5)
e) what is the domain of f
f) what is the range of f
5
6
7
8
9
Section 3.3 graphs of functions
#5 – 12, we will call each function graphed in problems 5-12 f(x). For what values of x is
a) f(x) ≥ 0 b) f(x) ≤0
5)
7).
6)
8)
3.3 graphs of functions
9)
10)
11)
12)
Section 3.3 graphs of functions
#13 – 23: Find the following:
a) the interval(s) where the function graphed is increasing
b) the interval(s) where the function graphed is decreasing
c) The values of x (if any) where the function has a local maximum
d) The local maximum value (if any)
e) The values of x (if any) where the function has a local minimum
f) The local minimum values (if any)
13)
14)
15)
16)
Section 3.3 graphs of functions
17)
19)
18)
Section 3.3 graphs of functions
20)
21)
23)
22)
Section 3.3 graphs of functions
24) Find the average rate of change of f(x) = (x-2)2 -4
a) from 1 to 2
b) from 3 to 5
25) Find the average rate of change of f(x) = (x-3)2 – 2
a) from 1 to 3
b) from 4 to 5
26) find the average rate of change of f(x) = x3 – 2x + 1
a) from -3 to -2
b) from -1 to 1
27) Find the average rate of change of f(x) = x3 – 3x2 + 5
a) from -3 to -2
b) from 4 to 6
28) The number of people P(t) (in hundreds) infected t days after an epidemic begins is approximated by
10ln(0.19𝑡 + 1)
𝑃(𝑡) =
0.19𝑡 + 1
The graph modeling the first 40 days of the epidemic is depicted below. The x-coordinate of each point
represents the number of days since the epidemic began and the y-coordinate the number of people (in
hundreds) infected.
a) Find the interval(s) where the graph is increasing and decreasing. Interpret the result.
b) What was the maximum number of people infected? When did it occur?
Section 3.3 graphs of functions
29) The function A(x) = 0.003631x3 -0.03746x2 + 0.1012x + 0.009
approximates blood alcohol concentration in a 170-lb woman x hours after drinking 2 ounces of alcohol
on an empty stomach. The graph of this function for the first 5 hours is drawn below. Where the xcoordinate of each point represents how long it has been since the woman had her drink and the ycoordinate represents her blood alcohol concentration.
a) Find the intervals where the graph is increasing and decreasing. Interpret the result.
b) What was the maximum blood alcohol concentration? When did it occur?
Section 3.3 graphs of functions
30) The percent of concentration of a drug in the bloodstream x hours after a drug is administered is
given by the function:
𝐾(𝑥) =
4𝑥
+ 27
3𝑥 2
The graph for the first 10 hours is drawn below.
a) Find the interval(s) where the graph is increasing and decreasing. Interpret the result>
b) What was the maximum concentration? When did this occur?
Section 3.4: Piecewise-defined functions
#1-6: Find the indicated value for each function.
1)
𝑓(𝑥) = {
a) f(-5)
3𝑥, 𝑖𝑓 𝑥 < 0
2𝑥 + 1, 𝑖𝑓 𝑥 ≥ 0
b) f(0)
c) f(2)
𝑥 − 5, 𝑖𝑓 𝑥 ≤ 5
2) 𝑓(𝑥) = {
2𝑥 − 4, 𝑖𝑓 𝑥 > 5
a) f(0)
b) f(5)
c) f(6)
𝑥 − 5, 𝑖𝑓 𝑥 < −1
𝑥, 𝑖𝑓 − 1 ≤ 𝑥 ≤ 2
3) 𝑔(𝑥) = {
𝑥 + 2, 𝑖𝑓 𝑥 > 2
a) g(-1)
b) g(2)
c) g(0)
2𝑥, 𝑖𝑓 𝑥 ≤ 0
4) g(x) = {𝑥, 𝑖𝑓 0 < 𝑥 ≤ 3
−5𝑥 𝑖𝑓 𝑥 > 3
a) g(0)
b) g(3)
c) g(-2)
𝑥 2 − 10, 𝑖𝑓 𝑥 < −10
5) 𝑘(𝑥) = {𝑥 2 , 𝑖𝑓 − 10 ≤ 𝑥 ≤ 10
𝑥 2 + 10. 𝑖𝑓 𝑥 > 10
a) k(-10)
b) k(11)
c) k(0)
2𝑥 2 − 3, 𝑖𝑓 𝑥 < 2
6) 𝑘(𝑥) = { 𝑥 2 , 𝑖𝑓 2 ≤ 𝑥 ≤ 4
5𝑥 − 7 𝑖𝑓 𝑥 > 4
a) k(2)
b) k(4)
c) k(5)
Section 3.4: Piecewise-defined functions
#7-12: sketch a graph of each function.
7)
𝑓(𝑥) = {
3𝑥, 𝑖𝑓 𝑥 < 0
2𝑥 + 1, 𝑖𝑓 𝑥 ≥ 0
𝑥 − 5, 𝑖𝑓 𝑥 ≤ 5
8) 𝑓(𝑥) = {
2𝑥 − 4, 𝑖𝑓 𝑥 > 5
𝑥 − 5, 𝑖𝑓 𝑥 < −1
𝑥, 𝑖𝑓 − 1 ≤ 𝑥 ≤ 2
9) 𝑔(𝑥) = {
𝑥 + 2, 𝑖𝑓 𝑥 > 2
2𝑥, 𝑖𝑓 𝑥 ≤ 0
10) g(x) = {𝑥 + 1, 𝑖𝑓 0 < 𝑥 ≤ 3
−5𝑥 𝑖𝑓 𝑥 > 3
𝑥 2 − 2, 𝑖𝑓 𝑥 < −1
11) 𝑘(𝑥) = {𝑥 2 , 𝑖𝑓 − 1 ≤ 𝑥 ≤ 1
𝑥 2 + 2. 𝑖𝑓 𝑥 > 1
2𝑥 2 − 3, 𝑖𝑓 𝑥 < 2
12) 𝑘(𝑥) = { 𝑥 2 , 𝑖𝑓 2 ≤ 𝑥 ≤ 4
5𝑥 − 7 𝑖𝑓 𝑥 > 4
13) In the 1995 tax form a tax rate schedule is given for people whose filing status is single. Part of the
table is shown below:
If the taxable income is over... But not over-- then the tax is... of the amount over-$0
$23,350
15%
$0
$23,350
$56,550 $3,502.50 + 28%
$23,350
$56,550
$117,950 $12,798.50 + 31%
$56,550
a. Write the defining rule for a piecewise function T(x) giving the tax owed by a person
whose taxable income is x, where x is less than $117,950.
b. Evaluate the function to find the tax owed by a single person whose taxable income in
1995 was $31,950.
Section 3.4 piecewise defined functions
14) In the 2005 tax form a tax rate schedule is given for people whose filing status is single. Part of the
table is shown below:
If the taxable income is over... But not over-- then the tax is... of the amount over-$0
$28,000
10%
$0
$28,000
$60,000
$2800 + 20%
$28,000
$60,000
$200,000
9200 + 25%
$60,000
a) Write the defining rule for a piecewise function T(x) giving the tax owed by a person
whose taxable income is x, where x is less than $200,000.
b) Evaluate the function to find the tax owed by a single person whose taxable income
in 2005 was $50,000.
15) Assume you work at a company where you are paid hourly.
You are paid $7.80 per hour for regular time (less than or equal to thirty-five hours)
and time and a half for overtime hours up to forty-five hours in one week.
If you are asked to work forty-five or more hours in one week you are paid double-time.
a) Write this information in the form of a piecewise-defined function
b) Use your formula to compute the pay for working 50 hours in a week.
16) Assume you work at a company where you are paid hourly.
You are paid $10.00 per hour for regular time (less than or equal to forty hours)
and time and a half for overtime hours up to fifty hours in one week.
If you are asked to work fifty or more hours in one week you are paid double-time.
a) Write this information in the form of a piecewise-defined function
b) Use your formula to compute the pay for working 50 hours in a week.
Section 3.5 Transformations of functions
The first few pages are the problems I will solve when I lecture. The problems labeled lecture are not
homework problems. Understanding these problems will help you complete the home work problems.
Lecture problem 1) Graph the function f(x) = x2 using a graphing calculator. Then graph each of the
following functions (using your calculator) and describe how the graph compares to the graph of f(x) = x2
.
a) f(x-3) = (x-3)2
b) f(x-4) = (x-4)2
c) f(x-6) = (x-6)2
Lecture problem 2) Graph the function f(x) = |𝑥| using a graphing calculator. Then graph each of the
following functions (using your calculator) and describe how the graph compares to the graph of f(x) =
|𝑥| .
a) f(x-2) = |𝑥 − 2|
b) 𝑓(𝑥 − 4) = |𝑥 − 4|
c) 𝑓(𝑥 − 6) = |𝑥 − 6|
Lecture problem 3) Graph the function f(x) = √𝑥 using a graphing calculator. Then graph each of the
following functions (using your calculator) and describe how the graph compares to the graph of f(x) =
√𝑥
a) 𝑓(𝑥 − 2) = √𝑥 − 2
b) 𝑓(𝑥 − 3) = √𝑥 − 3
Lecture problem 4) Describe how the graph of each function compares to that of a “common graph”
a) g(x) = (x-7)2
b) h(x) = √𝑥 − 11
c) f(x) = |𝑥 − 9|
Lecture problem 5) Graph the function f(x) = x2 using a graphing calculator. Then graph each of the
following functions (using your calculator) and describe how the graph compares to the graph of f(x) =
x2 .
a) f(x+3) = (x+3)2
b) f(x+4) = (x+4)2
Lecture problem 6) Graph the function f(x) = |𝑥| using a graphing calculator. Then graph each of the
following functions (using your calculator) and describe how the graph compares to the graph of f(x) =
|𝑥| .
a) f(x+2) = |𝑥 + 2|
b) 𝑓(𝑥 + 4) = |𝑥 + 4|
Section 3.5 Transformations of functions
Lecture problem 7) Graph the function f(x) = √𝑥 using a graphing calculator. Then graph each of the
following functions (using your calculator) and describe how the graph compares to the graph of f(x) =
√𝑥
a) 𝑓(𝑥 + 2) = √𝑥 + 2
b) 𝑓(𝑥 + 3) = √𝑥 + 3
Lecture problem 8) Describe how the graph of each function compares to that of a “common graph”
a) g(x) = (x+7)2
b) h(x) = √𝑥 + 11
c) f(x) = |𝑥 + 9|
Lecture problem 9) Graph the function f(x) = x2 using a graphing calculator. Then graph each of the
following functions (using your calculator) and describe how the graph compares to the graph of f(x) = x2
.
a) f(x)-3 = x2 -3
b) f(x) -4 = x2 - 4
Lecture problem 10) Graph the function f(x) = |𝑥| using a graphing calculator. Then graph each of the
following functions (using your calculator) and describe how the graph compares to the graph of f(x) =
|𝑥| .
a) f(x)-2 = |𝑥| − 2
b) 𝑓(𝑥) − 4 = |𝑥| − 4
Lecture problem 11) Graph the function f(x) = √𝑥 using a graphing calculator. Then graph each of the
following functions (using your calculator) and describe how the graph compares to the graph of f(x) =
√𝑥
a) 𝑓(𝑥) − 2 = √𝑥 − 2
b) 𝑓(𝑥) − 3 = √𝑥 − 3
Lecture problem 12) Describe how the graph of each function compares to that of a “common graph”
a) g(x) = x2 - 7
b) h(x) = √𝑥 − 11
c) f(x) = |𝑥| − 9
Lecture problem 13) Graph the function f(x) = x2 using a graphing calculator. Then graph each of the
following functions (using your calculator) and describe how the graph compares to the graph of f(x) = x2
.
a) f(x)+3 = x2 +3
b) f(x) +4 = x2 + 4
Section 3.5 Transformations of functions
Lecture problem 14) Graph the function f(x) = |𝑥| using a graphing calculator. Then graph each of the
following functions (using your calculator) and describe how the graph compares to the graph of f(x) =
|𝑥| .
a) f(x)+2 = |𝑥| + 2
b) 𝑓(𝑥) + 4 = |𝑥| + 4
Lecture problem 15) Graph the function f(x) = √𝑥 using a graphing calculator. Then graph each of the
following functions (using your calculator) and describe how the graph compares to the graph of f(x) =
√𝑥
a) 𝑓(𝑥) + 2 = √𝑥 + 2
b) 𝑓(𝑥) + 3 = √𝑥 + 3
Lecture problem 16) Describe how the graph of each function compares to that of a “common graph”
a) g(x) = x2 + 7
b) h(x) = √𝑥 + 11
c) f(x) = |𝑥| + 9
Lecture problem 17) let f(x) = x2 describe how the graph of –f(x) = -x2 compares to the graph of
f(x) = x2
Lecture problem 18) let f(x) = x3 describe how the graph of –f(x) = -x3 compares to the graph of f(x) = x3
Lecture problem 19) let f(x) = √𝑥 describe how the graph of –f(x) = −√𝑥 relates to the graph of f(x) =√𝑥
Lecture problem 20) let f(x) = |𝑥| describe how the graph of –f(x) = −|𝑥| relates to the graph of f(x) = |𝑥|
Lecture problem 21) let f(x) = √𝑥 describe how the graph of f(-x) = √−𝑥 relates to the graph of f(x) =
√𝑥
Lecture problem 22) let f(x) = √𝑥 + 2 describe how the graph of f(-x) = √−𝑥 + 2 relates to the graph of
𝑓(𝑥) = √𝑥 + 2
Lecture problem 23) let f(x) = √𝑥 + 3 describe how the graph of f(-x) = √−𝑥 + 3 relates to the graph of
𝑓(𝑥) = √𝑥 + 3
Lecture problem 24) let f(x) = √𝑥 − 5 describe how the graph of f(-x) = √−𝑥 − 5 relates to the graph of
𝑓(𝑥) = √𝑥 − 5
Section 3.5 Transformations of functions
THE HOMEWORK PROBLEMS FOR SECTION 3.5 START HERE.
#1-16 describe how the graph of the given function relates to the graph of a common function
1) f(x) = (x-2)2 + 5
2) f(x) = (x-5)2 + 2
3) f(x) = √𝑥 + 3 – 5
4) f(x) = √𝑥 − 1 − 4
5) 𝑓(𝑥) = |𝑥 − 3| + 1
6) 𝑓(𝑥) = |𝑥 − 5| + 4
7) f(x) = -(x-3)2 + 2
8) f(x) = -(x+1)2 – 4
9) 𝑓(𝑥) = −√𝑥 − 3 − 1
10) 𝑓(𝑥) = −√𝑥 + 2 − 1
11) 𝑓(𝑥) = √−𝑥 + 3
12) f(x) = √−𝑥 + 2
13) 𝑓(𝑥) = √−𝑥 + 1
14) f(x) = √−𝑥 + 4
15) f(x) = −√−𝑥 − 1
16) f(x) = −√−𝑥 + 2
17) Write the function whose graph has the same shape as the graph of f(x) = x2 but is shifted to the
right 3 units.
18) Write the function whose graph has the same shape as the graph of f(x) = x2 but is shifted to the left
2 units.
19) Write the function whose graph has the same shape as the graph of f(x) = x2 but is shifted to the
right 3 units and up 2 units.
20) Write the function whose graph has the same shape as the graph of f(x) = x2 but is shifted to the left
3 units and down 2 units.
21) Write the function whose graph has the same shape as the graph of f(x) = x2 but is reflected over the
x-axis and is shifted to the right 5 units and down 2 units.
22) Write the function whose graph has the same shape as the graph of f(x) = x2 but is reflected over the
x-axis and is shifted to the left 2 units and up 4 units.
23) Write the function whose graph has the same shape as the graph of f(x) = √𝑥 + 3 but is reflected
over the y-axis.
24) Write the function whose graph has the same shape as the graph of f(x) = √𝑥 + 5 but is reflected
over the y-axis.
25) Write the function whose graph has the same shape as the graph of f(x) = x2 but is reflected over the
x-axis and is shifted to the left 4 units.
26) Write the function whose graph has the same shape as the graph of f(x) = x2 but is reflected over the
x-axis and is shifted to the right 4 units.
Section 3.5 Transformations of functions
27) Write the function whose graph has the same shape as the graph of f(x) = x2 but is reflected over the
x-axis and is shifted to the left 5 units and up 2 units.
28 – 35: Each graph is a rigid transformation of a common function. Write the equation of the function.
29)
28)
31)
30)
Section 3.5 Transformations of functions
33)
32)
34) Graph the function f(x) = x2 using your graphing calculator. Then graph the following functions and
describe how the graphs relate.
1
a) g(x) = (3x)2 b) k(x) = (5x)2
c) 𝑛(𝑥) = (2 𝑥)
2
1
2
d) 𝑝(𝑥) = (4 𝑥)
35) Graph the function f(x) = |𝑥| using your graphing calculator. Then graph the following functions and
describe how the graphs relate.
a) 𝑘(𝑥) = |3𝑥|
b) 𝑔(𝑥) = |2𝑥|
1
1
c) j(x) = |2 𝑥|
d) l(x) = |4 𝑥|
36) Graph the function f(x) = x2 using your graphing calculator. Then graph the following functions and
describe how the graphs relate.
a) g(x) = 3x2
b) k(x) = 5x2
c) m(x) = - 10x2
1
2
d) 𝑛(𝑥) = 𝑥 2
1
e) 𝑝(𝑥) = − 4 𝑥 2
37) Graph the function f(x) = |𝑥| using your graphing calculator. Then graph the following functions and
describe how the graphs relate.
a) 𝑘(𝑥) = 3|𝑥|
b) 𝑔(𝑥) = 2|𝑥|
1
4
c) j(x) = − |𝑥|
1
4
d) l(x) = |𝑥|
Section 3.6: mathematical modeling – building functions
1) A campground owner has 800 meters of fencing. He wants to enclose a rectangular field. Let W
represent the width of the field. Follow these steps to find the dimensions of the field that yields the
largest area.
a) Write an expression for the length of the field
b) Write an equation for the area of the field.
c) Find the value of w leading to the maximum area
d) Find the value of L leading to the maximum area
e) Find the maximum area.
2) A campground owner has 1000 meters of fencing. He wants to enclose a rectangular field bordering
a river, with no fencing needed along the river. Let W represent the width of the field. Follow these
steps to find the dimensions of the field that yields the largest area.
a) Write an expression for the length of the field
b) Write an equation for the area of the field.
c) Find the value of w leading to the maximum area
d) Find the value of L leading to the maximum area
e) Find the maximum area.
3) A campground owner has 1400 meters of fencing. He wants to enclose a rectangular field bordering
a river, with no fencing needed along the river, and let W represent the width of the field. Follow these
steps to find the dimensions of the field that yields the largest area.
a) Write an expression for the length of the field
b) Write an equation for the area of the field.
c) Find the value of w leading to the maximum area
d) Find the value of L leading to the maximum area
e) Find the maximum area.
Section 3.5 Transformations of functions:
Here is a summary of the rules presented in this section.
Up and down shifts
y = f(x) + k (k>0)
Y = f(x) – k (k>0)
Left and right shifts
y = f(x+h) (h >0)
y = f(x-h) (h>0)
Raise the graph of f(x) by k units
Lower the graph of f(x) by k units
Shift graph of f(x) left h units
Shift graph of f(x) right h units
Reflections
y= -f(x)
y= f(-x)
Compressing and stretching
y = af(x) (a >0)
Compressing and stretching
y = f(ax) (a >0)
Reflects graph of f(x) about x-axis
Reflects graph of f(x) about y-axis
Stretches the graph of f vertically if a >1
Compress graph of f vertically if 0<a<1
Stretches the graph of f horizontally if 0 <a<1
Compresses the graph of f horizontally if a>1
Section 3.6: mathematical modeling – building functions
4) A campground owner has 1000 meters of fencing. He wants to enclose a rectangular field bordering
a river, with no fencing needed along the river, and let W represent the width of the field.
a) Write an expression for the length of the field
b) Write an equation for the area of the field.
c) Find the value of w leading to the maximum area
d) Find the value of L leading to the maximum area
e) Find the maximum area.
5) A fence must be built to enclose a rectangular area of 20,000 square feet. Fencing material costs
$2.50 per foot for the two sides facing north and south (call these sides the length, and $3.20 per foot
for the other two sides (call these sides the width). Follow these steps to find the cost of the least
expensive fence.
a) Write an equation for the length of the field.
b) Write an equation for the cost of the field.
c) Find the value of W leading to the minimum cost
d) Find the value of L leading to the minimum cost
e) Find the minimum cost.
6) A fence must be built to enclose a rectangular area of 20,000 square feet. Fencing material costs
$2.00 per foot for the two sides facing north and south (call these sides the length, and $4.00 per foot
for the other two sides (call these sides the width). Follow these steps to find the cost of the least
expensive fence.
a) Write an equation for the length of the field.
b) Write an equation for the cost of the field.
c) Find the value of W leading to the minimum cost
d) Find the value of L leading to the minimum cost
e) Find the minimum cost.
Section 3.6: mathematical modeling – building functions
7) A fence must be built in a large field to enclose a rectangular area of 25,600 square meters. One side
of the area is bounded by an existing fence; no fence is needed there. Material for the fence costs $3.00
per meter for the two ends, and $1.50 per meter for the side opposite the existing fence. Find the cost
of the least expensive fence.
a) Write an equation for the length of the field.
b) Write an equation for the cost of the field.
c) Find the value of W leading to the minimum cost
d) Find the value of L leading to the minimum cost
e) Find the minimum cost.
8) A fence must be built in a large field to enclose a rectangular area of 10,000 square meters. One side
of the area is bounded by an existing fence; no fence is needed there. Material for the fence costs $5.00
per meter for the two ends, and $2.00 per meter for the side opposite the existing fence. Find the cost
of the least expensive fence.
a) Write an equation for the length of the field.
b) Write an equation for the cost of the field.
c) Find the value of W leading to the minimum cost (round to 2 decimals)
d) Find the value of L leading to the minimum cost (round to 2 decimals)
e) Find the minimum cost.
9) An open box with a square base is to be made from a square piece of cardboard 10 inches on a side
by cutting out a square ( x inches by x inches) from each corner and turning up the sides. (round to 2
decimals if needed)
a) Sketch a diagram that models the problem.
b) Write an equation for the volume of the box.
c) Graph the volume function using your graphing calculator and find the value of x that makes V the
largest.
Section 3.6: mathematical modeling – building functions
10) An open box with a square base is to be made from a square piece of cardboard 12 inches on a side
by cutting out a square ( x inches by x inches) from each corner and turning up the sides.
a) Sketch a diagram that models the problem.
b) Write an equation for the volume of the box.
c) Graph the volume function using your graphing calculator and find the value of x that makes V the
largest.
11) An open box is to be made by cutting a square corner of a 20 inch by 20 inch piece of metal then
folding up the sides. What size square should be cut from each corner to maximize volume? (round to 2
decimals if needed)
a) Sketch a diagram that models the problem.
b) Write an equation for the volume of the box.
c) Graph the volume function using your graphing calculator and find the value of x that makes V the
largest. (round to 2 decimal places if needed)
12) An open box is to be made by cutting a square corner of a 30 inch by 30 inch piece of metal then
folding up the sides. What size square should be cut from each corner to maximize volume?
a) Sketch a diagram that models the problem.
b) Write an equation for the volume of the box.
c) Graph the volume function using your graphing calculator and find the value of x that makes V the
largest. (round to 2 decimal places if needed)
Chapter 3 Review
#1-4: Determine whether the equation defines y as a function of x. Hint, solve the equation for y and
sketch a graph using your calculator, then solve like problems #7-12.
1) y = (x-3)2 +4
2) x = y2 + 4
3) (x-3)2 + y2 = 9
4) 𝑦 = √𝑥 − 1
#5 – 8: Determine the domain and range of each function, write your answer in interval notation when
appropriate.
5)
6)
7)
8)
Chapter 3 Review
#9-11: Use algebra to find the domain of each function. Write your answer in interval notation
𝑥−4
9) 𝑓(𝑥) = 𝑥 2 +6𝑥−7
10) 𝑓(𝑥) = √𝑥 + 5
11) f(x) = 2x – 6
#12 – 15: let f(x) = x2 + 3x – 4 and g(x) = 7x – 7, find the following
12) (f-g)(x)
13) (𝑓 ∘ 𝑔)(𝑥)
14) (f+g)(5)
#16 – 17: Find the difference quotient; that is find
16) f(x) = 2x – 3
15) (𝑔 ∘ 𝑓)(4)
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
17) f(x) = x2 + 5x + 2
Use the graph below to answer #18 – 23, call the function graphed below h(x)
18)
19)
20)
21)
22)
23)
find the x-intercepts
find the y-intercept
for what values of x is h(x) = -4
find h(-4)
what is the domain of h
what is the range of h
Chapter 3 review
#24 – 25, we will call each function graphed in problems 5-12 f(x). For what values of x is
a) f(x) ≥ 0 b) f(x) ≤0
24)
25)
#26 – 27: Find the following:
a) the interval(s) where the function graphed is increasing
b) the interval(s) where the function graphed is decreasing
c) The values of x (if any) where the function has a local maximum
d) The local maximum value (if any)
e) The values of x (if any) where the function has a local minimum
f) The local minimum values (if any)
26)
27)
Chapter 3 review
28) Find the average rate of change of f(x) = (3x-2)2 -4
from 4 to 5
29) Find the average rate of change of f(x) = x3 + 6x2 from 0 to 2
#30-31: sketch a graph of each function.
30)
3𝑥 + 1, 𝑖𝑓 𝑥 < 2
𝑓(𝑥) = {
𝑥 − 1, 𝑖𝑓 𝑥 ≥ 2
𝑥 − 5, 𝑖𝑓 𝑥 ≤ 5
31) 𝑓(𝑥) = {2𝑥 − 4, 𝑖𝑓 5 < 𝑥 ≤ 7
2𝑥, 𝑖𝑓 𝑥 > 7
#31-35 describe how the graph of the given function relates to the graph of a common function
31) f(x) = (x+2)2 - 4
32) f(x) = −√𝑥 − 3 – 5
34) 𝑓(𝑥) = √−𝑥 + 2
35) f(x) = -(x-3)3
33) 𝑓(𝑥) = |𝑥 − 3| + 4
36) Write the function whose graph has the same shape as the graph of f(x) = x2 but is shifted to the left
3 units and down 2 units.
37) Write the function whose graph has the same shape as the graph of f(x) = √𝑥 but is reflected over
the y-axis.
38) Write the function whose graph has the same shape as the graph of f(x) = x2 but is reflected over the
x-axis and is shifted to the left 4 units and up 2 units.
39) A campground owner has 2000 feet of fencing. He wants to enclose a rectangular field bordering a
river. Let W represent the width of the field. Follow these steps to find the dimensions of the field that
yields the largest area. (round all answers to 2 decimal places if needed)
a) Write an expression for the length of the field
b) Write an equation for the area of the field.
c) Find the value of w leading to the maximum area
d) Find the value of L leading to the maximum area
e) Find the maximum area
Chapter 3 Review
40) A fence must be built in a large field to enclose a rectangular area of 1,000 square meters. One side
of the area is bounded by an existing fence; no fence is needed there. Material for the fence costs
$10.00 per meter for the two ends, and $6.00 per meter for the side opposite the existing fence. Find
the cost of the least expensive fence. (round all answers to 2 decimal places if needed)
a) Write an equation for the length of the field.
b) Write an equation for the cost of the field.
c) Find the value of W leading to the minimum cost
d) Find the value of L leading to the minimum cost
e) Find the minimum cost.